Complete classification for simple root cyclic codes over local rings $\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$

Abstract: Let $p$ be a prime integer, $n,s\geq 2$ be integers satisfying ${\rm gcd}(p,n)=1$, and denote $$R=\mathbb{Z}<em>{p^{s}[v]/\langle} v<sup>2-pv\rangle$$. Then $R$ is a local non-principal ideal ring of $p<sup>{2s}$ elements. First, the structure of any cyclic code over $R$ of length $n$ and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over $R$ of length $n$ are investigated. Finally, we list some optimal $2$-quasi-cyclic self-dual linear codes over $\mathbb{Z}_4$ of length $30$ and extremal $4$-quasi-cyclic self-dual binary linear $[60,30,12]$ codes derived from cyclic codes over $\mathbb{Z}</em>{4}[v]/\langle v<sup>2+2v\rangle$ of length $15$.